Extension of the Monge point beyond simplices

Determine whether the Monge point, which is defined for n-simplices, can be extended to polytopes that are not simplices, and characterize such an extension so that its relationship to the point P_1 is made precise when applicable.

Background

Within the definition of P_λ points, the paper notes that when a polytope σ is orthocentric, the point P_1 coincides with the Monge point defined for simplices. However, outside the class of simplices, the existence or definition of a Monge-type point has not been addressed in the literature, leading the author to refer to P_1 as a quasi-orthocenter for general polytopes.

This raises a natural question about whether the Monge point can be generalized beyond simplices to broader classes of polytopes, potentially aligning with the framework of P_λ points presented in the paper. Establishing such an extension would clarify the role of P_1 in non-simplicial settings and unify orthocentric and non-orthocentric cases.

References

If σ is not orthocentric, one can define the notion of a "Monge Point" (see [Buba]) in the simplex. This point coincides with P_1 when σ is orthocentric; the extension of this notion beyond the simplex has not been studied, hence we call P_1 the quasi-orthocenter.

— Cycling Along Euler Road  (2509.10460 - Wyrzykowski, 4 Aug 2025) in Section 2, Definition of P_λ points (footnote)